Finite morphisms of schemes

Depends on

• Affine morphisms
• Subalgebra generated by a subset, algebras of finite type, and module-finite algebras

Used by

• Every smooth proper curve admits a projective embedding Corollary
• Finite morphisms are proper Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• The genus relation for unramified covers of curves Corollary
• Frobenius on the affine line is finite flat but not smooth Counterexample
• Normalization and blowup are different operations Counterexample
• Normalization of a non-normal surface is not a point blowup Counterexample
• Degree of a nonconstant morphism of curves Definition
• Geometric fibre functor and étale fundamental group Definition
• Geometric genus of a singular curve Definition
• Ramification points, branch points and unramifiedness Definition
• The different divisor of a generically separable morphism of curves Definition
• Closed immersion from a quotient ring Example
• Finite field extensions and etaleness Example
• Finite power map of the affine line Example
• Ramification of the double cover y²=f(x) Example
• Riemann-Hurwitz for a tame double cover with 2r branch points Example
• The empty morphism is finite, proper and projective Example
• Under AC, effective divisors on normal proper curves give finite subschemes of the same degree Example
• Blowing up a non-regular point strictly increases the finite normalization subalgebra Lemma
• Closed immersions are proper Lemma
• Composite of a finite morphism and a proper morphism is proper Lemma
• Fibre degree of the finite locally free map to the projective line Lemma
• Finite decomposition around isolated fibre points after an elementary étale change Lemma
• Finite étale algebras have finite locally free underlying modules Lemma
• Finite is affine and local on its target Lemma
• Finite morphisms survive base change and composition Lemma
• Finite neighbourhood of an isolated fibre point after elementary etale change Lemma
• Finite pullback preserves absolute ampleness Lemma
• Finite schemes over projective schemes are projective over a Noetherian affine base Lemma
• Finite-fibre and pointwise characterizations of quasi-finiteness Lemma
• Local support and index bound for the different of a curve map Lemma
• Normalization is unchanged under finite birational maps of reduced curves Lemma
• Proper normal curve rational function map Lemma
• Scheme Zariski Main factorization for separated quasi-finite morphisms Lemma
• The finite normalization of a curve factors through the blowup of a closed point Lemma
• The number of contracted curves drops by one after factoring through a point blowup Lemma
• The twisting sheaf of projective space is very ample and ample Lemma
• A proper quasi-finite morphism is finite Theorem
• Canonical bundle formula with the different Theorem

…and 9 more results.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Stacks Project, Morphisms of Schemes §29.45 (standard reference, not scraped)
• Vakil, The Rising Sea §§8.3, 11.3, 17.4 (standard reference, not scraped)